Chengchun Hao

Preprints

1. Global well-posedness for the free-boundary incompressible viscous resistive MHD in critical Besov spaces (with Wei Zhang, Jie Fu and Siqi Yang)
   Preprint, 2024.
   This paper is devoted to proving the global well-posedness of the free boundary problem for the incompressible viscous resistive MHD equations. Under the Lagrangian coordinates, there exists a unique global solution in the half-space if the norm of the initial data in critical homogeneous Besov space $\dot{B}_{p, 1}^{-1+N / p}(\mathbb{R}_{+}^N)$ is small enough for $p \in[N,2N-1)$. Inspired by the results in (J. Funct. Anal. 256: 881--927, 2009) and (J. Differ. Equations 274: 613--651, 2021) in half-space, we establish the maximal $L^{1}$-regularity for both the Stokes equations without surface stress and the linearized equations of the magnetic field with zero boundary condition. The existence and uniqueness of the solutions of nonlinear problems is proved by using the Banach contraction mapping principle.
2. Global well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximation (with Wei Zhang, Jie Fu and Siqi Yang)
   Preprint, 2024.
   This paper focuses on the global well-posedness of the Oberbeck-Boussinesq approximation for the unsteady motion of a drop in another bounded fluid separated by a closed interface with surface tension. We assume that the initial states of the drop are close to a ball $B_{R}$ with the same volume as the drop, and that the boundary of the drop is a small perturbation of the boundary of $B_{R}$. To begin, we introduce the Hanzawa transformation with an added barycenter point to obtain the linearized Oberbeck-Boussinesq approximation in a fixed domain. Then, we establish time-weighted estimates of solutions for the shifted equation using maximal $L^p$-$L^q$ regularities for the two-phase fluid motion of the linearized system, as obtained by Hao and Zhang in 2022. Using the time decay estimates of the semigroup, we obtain decay time-weighted estimates of solutions for the linearized problem. Additionally, we prove that these estimates are less than the sum of the initial value and its own square and cube by estimating the corresponding non-linear terms. Finally, the existence and uniqueness of solutions in the finite time interval $(0,T)$ was proven by Hao and Zhang in 2023. After that, we demonstrate that the solutions can be extended beyond $T$ by analyzing the properties of the roots of algebraic equations.
3. On the motion of the closed free surface in three-dimensional incompressible ideal MHD with surface tension (with Siqi Yang)
   Preprint, 2023.   
   We develop the local regularity theory for the three-dimensional free boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension, and we consider the relation between the geometric characteristics of the free boundary and the regularity loss of a smooth solution. Our main results state that a smooth solution will preserve its initial regularity and remain smooth if certain conditions are met. These conditions include low regularity of the free boundary (curvature and normal velocity), the fluid velocity and magnetic fields, and the free boundary not self-intersecting. This result holds in a general bounded domain with closed surface without assuming the free boundary being a graph, and the regularity estimates are driven by the curvature bound and the pressure, which rely on the initial mean curvature. Our lower-order regularity estimates extend the a priori estimates for the flat domain $\mathbb{T}^2\times (0,1)$ in [C. Luo and J. Zhang, SIAM J. Math. Anal. 53 (2) (2021) 2595-2630] nontrivially, since the local coordinate can not preserve the curvature and the curvature of the flat domain vanishes. Moreover, we remove the requirement of the velocity on the initial boundary since the ultimate estimate does not depend on this quantity. This approach also avoids dealing with the spatial regularity of the flow map in Lagrangian coordinates, which may lack the maximal regularity and the geometric characteristics of the evolutionary domain. To our best knowledge, this is the first result involving the blow-up of the free boundary incompressible ideal MHD equations with surface tension.

Accepted Papers

1. A blow-up criterion for free boundary problem of incompressible neo-Hookean elastodynamics (with Jie Fu, Siqi Yang and Wei Zhang)
   SCIENTIA SINICA Mathematica, 55: 1–16, 2025.
   We prove a Beale-Kato-Majda type blow-up criterion for the solution to the free boundary problem of the three-dimensional incompressible neo-Hookean elastodynamics model. The result indicates that, under the Taylor-type sign condition, as long as some norms of the curl of the deformation tensor and the velocity field, the second fundamental form of the free boundary, the injective radius of the normal exponential map, and pressure remain bounded, and the gradients of the deformation tensor and the velocity field, together with the material derivatives of the gradient of pressure remain bounded on the free boundary, the solution can be continuously extended.



Published Journal Articles

■ Free boundary problems in fluid dynamics equations

2. Splash singularity for the free boundary incompressible viscous MHD (with Siqi Yang)
   Journal of Differential Equations, 379, 26–103, 2024.            
   In this paper, we prove the existence of smooth initial data for the two-dimensional free boundary incompressible viscous magnetohydrodynamics (MHD) equations, for which the interface remains regular but collapses into a splash singularity (self-intersects in at least one point) in finite time. The existence of the splash singularities is guaranteed by a local existence theorem, in which we need suitable spaces for the modified magnetic field together with the modification of the velocity and pressure such that the modified initial velocity is zero, and a stability result which allows us to construct a class of initial velocities and domains for an arbitrary initial magnetic field. It turns out that the presence of the magnetic field does not prevent the viscous fluid from forming splash singularities for certain smooth initial data.
3. A Beale-Kato-Majda criterion for free boundary incompressible ideal magnetohydrodynamics (with Jie Fu, Siqi Yang and Wei Zhang)
   Journal of Mathematical Physics 64(9): 091505, 16pp, 2023.          
   We prove a continuation criterion for the free boundary problem of three-dimensional incompressible ideal magnetohydrodynamic (MHD) equations in a bounded domain, which is analogous to the theorem given in [Beale, Kato and Majda 1984 Comm. Math. Phys. 94 61--66]. We combine the energy estimates of our previous works [Hao and Luo 2014 {Arch. Ration. Mech. Anal. 212 805--847] on incompressible ideal MHD and some analogous estimates in [Ginsberg 2021 SIAM J. Math. Anal.53(3) 3366--3384] to show that the solution can be continued as long as the curls of the magnetic field and velocity, the second fundamental form and injectivity radius of the free boundary and some norms of the pressure remain bounded, provided that the Taylor-type sign condition holds.
4. Local well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximation (with Wei Zhang)
   Communications on Pure and Applied Analysis 22(7): 2099-2131, 2023.     
   This paper is concerned with the local well-posedness of the Oberbeck-Boussinesq approximation for the unsteady motion of a drop in another fluid separated by a closed interface with surface tension. We devote to getting the linearized Oberbeck-Boussinesq approximation in the fixed domain by using the Hanzawa transformation, and using maximal $L^{p}$-$L^{q}$ regularities for the two-phase fluid motion of the linearized system obtained by the authors in [11] to establish the existence and uniqueness of the solutions of nonlinear problem with the help of the contraction mapping principle, in which the differences of nonlinear terms are estimated.
5. A priori estimates for free boundary problem of 3D incompressible inviscid rotating Boussinesq equations (with Wei Zhang)
   Zeitschrift für angewandte Mathematik und Physik  74(2): No. 80, 21pp, 2023.      
   In this paper, we consider the three-dimensional rotating Boussinesq equations (the “primitive” equations of geophysical fluid flows). Inspired by Christodoulou and Lindblad (Commu. Pure Appl. Math. 53:1536–1602, 2000), we establish a priori estimates of Sobolev norms for free boundary problem of inviscid rotating Boussinesq equations under the Taylor-type sign condition on the initial free boundary. Using the same method, we can also obtain a priori estimates for the incompressible inviscid rotating MHD system with damping.
   @Article{HZ2023,
   author = {Chengchun Hao and Wei Zhang},
   journal = {Zeitschrift f\"ur angewandte Mathematik und Physik},
   title = {A priori estimates for free boundary problem of 3D incompressible inviscid rotating {B}oussinesq equations},
   year = {2023},
   month = {mar},
   number = {2},
   volume = {74},
   doi = {10.1007/s00033-023-01974-2}, }
6. Maximal $L^p$-$L^q$ regularity for two-phase fluid motion in the linearized Oberbeck-Boussinesq approximation (with Wei Zhang)
   Journal of Differential Equations  322, 101-134, 2022.        
   This paper is concerned with the generalized resolvent estimate and the maximal $L^p$-$L^q$ regularity of the linearized Oberbeck-Boussinesq approximation for unsteady motion of a drop in another fluid without surface tension, which is indispensable for establishing the well-posedness of the Oberbeck-Boussinesq approximation for the two incompressible liquids separated by a closed interface. We prove the existence of $\mathcal{R}$-bounded solution operators for the model problems and the maximal $L^p$-$L^q$ regularity for the system. The key step is to prove the maximal $L^p$-$L^q$ regularity theorem for the linearized heat equation with the help of the $\mathcal{R}$-bounded solution operators for the corresponding resolvent problem and the Weis operator-valued Fourier multiplier theorem.
   @article {HZ22jde,
AUTHOR = {Hao, Chengchun and Zhang, Wei},
TITLE = {Maximal {$L^p$}-{$L^q$} regularity for two-phase fluid motion in the linearized {O}berbeck-{B}oussinesq approximation},
JOURNAL = {J. Differential Equations},
FJOURNAL = {Journal of Differential Equations},
VOLUME = {322},
YEAR = {2022},
PAGES = {101--134},
ISSN = {0022-0396},
MRCLASS = {35Q35 (76T06)},
MRNUMBER = {4398420},
DOI = {10.1016/j.jde.2022.03.022},
URL = {https://doi.org/10.1016/j.jde.2022.03.022},
}
7. Some results on free boundary problems of incompressible ideal magnetohydrodynamics equations (with Tao Luo)
   Electronic Research Archive 30(2), 404-424, 2022.        
   We survey some recent results related to free boundary problems of incompressible ideal magnetohydrodynamics equations, and present the main ideas in the proofs of the ill-posedness in 2D when the Taylor sign condition is violated given [1], and the well-posedness of a linearized problem given in [2] in general n-dimensions $(n\geqslant 2)$ when the Taylor sign condition is satisfied and the free boundaries are closed.
   @article {HL22era,
   AUTHOR = {Hao, Chengchun and Luo, Tao},
   TITLE = {Some results on free boundary problems of incompressible ideal magnetohydrodynamics equations},
   JOURNAL = {Electron. Res. Arch.},
   FJOURNAL = {Electronic Research Archive},
   VOLUME = {30},
   YEAR = {2022},
   NUMBER = {2},
   PAGES = {404--424},
   MRCLASS = {35Q35 (76W05)},
   MRNUMBER = {4399918},
   DOI = {10.3934/era.2022021},
   URL = {https://doi.org/10.3934/era.2022021},
   }
8. Well-posedness for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations (with Tao Luo)
   Journal of Differential Equations 299, 542-601, 2021.        
   The well-posedness theory is studied for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations in a bounded domain. We express the magnetic field in terms of the velocity field and the deformation tensors in Lagrangian coordinates, and substitute it into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, the velocity equation is linearized with respect to the position vector field whose time derivative is the velocity. In this formulation, a key idea is to use the Lie derivative of the magnetic field taking the advantage that the magnetic field is tangential to the free boundary and divergence free. This paper contributes to the program of developing geometric approaches to study the well-posedness of free boundary problems of ideal magnetohydrodynamics equations under the condition of Taylor sign type for general free boundaries not restricted to graphs.
   @article {HL21jde,
   AUTHOR = {Hao, Chengchun and Luo, Tao},
   TITLE = {Well-posedness for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations},
   JOURNAL = {J. Differential Equations},
   FJOURNAL = {Journal of Differential Equations},
   VOLUME = {299},
   YEAR = {2021},
   PAGES = {542--601},
   ISSN = {0022-0396},
   MRCLASS = {35Q35 (35R35 76W05)},
   MRNUMBER = {4295166},
   DOI = {10.1016/j.jde.2021.07.030},
   URL = {https://doi.org/10.1016/j.jde.2021.07.030},
   }
9. Ill-posedness of free boundary problem of the incompressible ideal MHD (with Tao Luo)
   Commun. Math. Phys. 376(1), 259–286, 2020.        
   In the present paper, we show the ill-posedness of the free boundary problem of the incompressible ideal magnetohydrodynamics (MHD) equations in two spatial dimensions for any positive vacuum permeability $\mu _0$, in Sobolev spaces. The analysis is uniform for any $\mu_0>0$.
   @article {HL20cmp,
   AUTHOR = {Hao, Chengchun and Luo, Tao},
   TITLE = {Ill-posedness of free boundary problem of the incompressible ideal {MHD}},
   JOURNAL = {Comm. Math. Phys.},
   FJOURNAL = {Communications in Mathematical Physics},
   VOLUME = {376},
   YEAR = {2020},
   NUMBER = {1},
   PAGES = {259--286},
   ISSN = {0010-3616},
   MRCLASS = {35R35 (35B35 35Q35)},
   MRNUMBER = {4093862},
   DOI = {10.1007/s00220-019-03614-1},
   URL = {https://doi.org/10.1007/s00220-019-03614-1},
   }
10. On the motion of free interface in ideal incompressible MHD
    Arch. Rational Mech. Anal. 224(2), 515–553, 2017.        
    For the free boundary problem of the plasma–vacuum interface to 3D ideal incompressible magnetohydrodynamics, the a priori estimates of smooth solutions are proved in Sobolev norms by adopting a geometrical point of view and some quantities such as the second fundamental form and the velocity of the free interface are estimated. In the vacuum region, the magnetic fields are described by the div–curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic fields are tangential to the interface, but we do not need any restrictions on the size of the magnetic fields on the free interface. We introduce the “fictitious particle” endowed with a fictitious velocity field in vacuum to reformulate the problem to a fixed boundary problem under the Lagrangian coordinates. The $L^2$-norms of any order covariant derivatives of the magnetic fields both in vacuum and on the boundaries are bounded in terms of initial data and the second fundamental forms of the free interface and the rigid wall. The estimates of the curl of the electric fields in vacuum are also obtained, which are also indispensable in elliptic estimates.
    @article {H17arma,
    AUTHOR = {Hao, Chengchun},
    TITLE = {On the motion of free interface in ideal incompressible {MHD}},
    JOURNAL = {Arch. Ration. Mech. Anal.},
    FJOURNAL = {Archive for Rational Mechanics and Analysis},
    VOLUME = {224},
    YEAR = {2017},
    NUMBER = {2},
    PAGES = {515--553},
    ISSN = {0003-9527},
    MRCLASS = {76W05 (35Q35 35R35)},
    MRNUMBER = {3614754},
    DOI = {10.1007/s00205-017-1082-7},
    URL = {https://doi.org/10.1007/s00205-017-1082-7},
    }
11. A priori estimates for the free boundary problem of incompressible neo-Hookean elastodynamics (with Dehua Wang)
    Journal of Differential Equations  261(1), 712–737, 2016.         
    A free boundary problem for the incompressible neo-Hookean elastodynamics is studied in two and three spatial dimensions. The a priori estimates in Sobolev norms of solutions with the physical vacuum condition are established through a geometrical point of view of Christodoulou and Lindblad (2000) [3]. Some estimates on the second fundamental form and velocity of the free surface are also obtained.
    @article {HW16jde,
    AUTHOR = {Hao, Chengchun and Wang, Dehua},
    TITLE = {A priori estimates for the free boundary problem of incompressible neo-{H}ookean elastodynamics},
    JOURNAL = {J. Differential Equations},
    FJOURNAL = {Journal of Differential Equations},
    VOLUME = {261},
    YEAR = {2016},
    NUMBER = {1},
    PAGES = {712--737},
    ISSN = {0022-0396},
    MRCLASS = {35R35 (35A01 35A08 35B45 35Q35 74J30 76A10 76D03)},
    MRNUMBER = {3487273},
    DOI = {10.1016/j.jde.2016.03.025},
    URL = {https://doi.org/10.1016/j.jde.2016.03.025},
    }
12. Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities
    Discrete Contin. Dyn. Syst. Ser. B  20(9), 2885-2931, 2015.        
    In this paper, we establish a priori estimates for three-dimensional compressible Euler equations with the moving physical vacuum boundary, the $\gamma$-gas law equation of state for $\gamma=2$ and the general initial density $\rho_0\in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which plays a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$.
    @article {H15dcds,
    AUTHOR = {Hao, Chengchun},
    TITLE = {Remarks on the free boundary problem of compressible {E}uler equations in physical vacuum with general initial densities},
    JOURNAL = {Discrete Contin. Dyn. Syst. Ser. B},
    FJOURNAL = {Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences},
    VOLUME = {20},
    YEAR = {2015},
    NUMBER = {9},
    PAGES = {2885--2931},
    ISSN = {1531-3492},
    MRCLASS = {35R35 (35Q31 35Q35)},
    MRNUMBER = {3402676},
    DOI = {10.3934/dcdsb.2015.20.2885},
    URL = {https://doi.org/10.3934/dcdsb.2015.20.2885},
    }
13. A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows (with Tao Luo)
    Arch. Rational Mech. Anal. 212(3), 805-847, 2014.         
    In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid magnetohydrodynamics equations in all physical spatial dimensions $n = 2$ and $3$ by adopting a geometrical point of view used in Christodoulou and Lindblad (Commun Pure Appl Math 53:1536–1602, 2000), and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids.
    @article {HL14arma,
    AUTHOR = {Hao, Chengchun and Luo, Tao},
    TITLE = {A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows},
    JOURNAL = {Arch. Ration. Mech. Anal.},
    FJOURNAL = {Archive for Rational Mechanics and Analysis},
    VOLUME = {212},
    YEAR = {2014},
    NUMBER = {3},
    PAGES = {805--847},
    ISSN = {0003-9527},
    MRCLASS = {76W05 (35B45 35Q35 35R35)},
    MRNUMBER = {3187678},
    DOI = {10.1007/s00205-013-0718-5},
    URL = {https://doi.org/10.1007/s00205-013-0718-5},
    }



■ Well-posedness of Cauchy problems to fluid dynamics equations

14. Well-posedness for a multidimensional viscous liquid-gas two-phase flow model (with Hai-Liang Li)
    SIAM J. Math. Anal. 44(3), 1304–1332, 2012.        
    The Cauchy problem of a multidimensional ($d\geqslant 2$) compressible viscous liquid-gas two-phase flow model is studied in this paper. We investigate the global existence and uniqueness of the strong solution for the initial data close to a stable equilibrium and the local-in-time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. A continuation criterion is also obtained for the local solution.
    @article {HL12sima,
    AUTHOR = {Hao, Chengchun and Li, Hai-Liang},
    TITLE = {Well-posedness for a multidimensional viscous liquid-gas two-phase flow model},
    JOURNAL = {SIAM J. Math. Anal.},
    FJOURNAL = {SIAM Journal on Mathematical Analysis},
    VOLUME = {44},
    YEAR = {2012},
    NUMBER = {3},
    PAGES = {1304--1332},
    ISSN = {0036-1410},
    MRCLASS = {76T10 (35A01 35A02 35B30 35D35 35Q35)},
    MRNUMBER = {2982713},
    DOI = {10.1137/110851602},
    URL = {https://doi.org/10.1137/110851602},
    }
15. Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces
    Zeitschrift für angewandte Mathematik und Physik 63(5), 825-834, 2012.        
    We investigate the Cauchy problem of a multidimensional chemotaxis model with initial data in critical Besov spaces. The global existence and uniqueness of the strong solution is shown for initial data close to a constant equilibrium state.
    @article {H12zamp,
    AUTHOR = {Hao, Chengchun},
    TITLE = {Global well-posedness for a multidimensional chemotaxis model in critical {B}esov spaces},
    JOURNAL = {Z. Angew. Math. Phys.},
    FJOURNAL = {Zeitschrift f\"{u}r Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Math\'{e}matiques et de Physique Appliqu\'{e}es},
    VOLUME = {63},
    YEAR = {2012},
    NUMBER = {5},
    PAGES = {825--834},
    ISSN = {0044-2275},
    MRCLASS = {35K45 (35A01 35A02 35B30 35D35 35Q92 92C17)},
    MRNUMBER = {2991216},
    DOI = {10.1007/s00033-012-0193-0},
    URL = {https://doi.org/10.1007/s00033-012-0193-0},
    }
16. Global well-posedness of compressible bipolar Navier-Stokes-Poisson equations (with Yiquan Lin, Hai-Liang Li)
    Acta Mathematica Sinica, English Series 28(5), 925-940, 2012.      
    We consider the initial value problem for multi-dimensional bipolar compressible Navier-Stokes-Poisson equations, and show the global existence and uniqueness of the strong solution in hybrid Besov spaces with the initial data close to an equilibrium state.
17. Well-posedness for the viscous rotating shallow water equations with friction terms
    J. Math. Phys. 52(2), 023101, 12pp, 2011.      
    We consider the Cauchy problem for viscous rotating shallow water equations with friction terms. The global existence of the solution in some hybrid spaces is shown for the initial data close to a constant equilibrium state away from the vacuum.
18. Well-posedness to the compressible viscous magnetohydrodynamic system
    Nonlinear Analysis: Real World Applications 12(6), 2962–2972, 2011.      
    This paper is concerned with the Cauchy problem of the compressible viscous magnetohydrodynamic (MHD) system in whole spatial space $R^d$ for $d\geqslant 3$. It is shown that the global solution exists uniquely in hybrid Besov spaces provided the initial data close to a constant equilibrium state away from the vacuum.
19. Cauchy problem for viscous shallow water equations with surface tension
    Discrete Contin. Dyn. Syst. Ser. B 13(3), 593--608, 2010.      
    We are concerned with the Cauchy problem for a viscous shallow water system with a third-order surface-tension term. The global existence and uniqueness of the solution in the space of Besov type is shown for the initial data close to a constant equilibrium state away from the vacuum by using the Friedrich's regularization and compactness arguments.
20. Cauchy problem for viscous rotating shallow water equations (with Ling Hsiao, Hai-Liang Li)
    Journal of Differential Equations 247(12), 3234–3257, 2009.      
    We consider the Cauchy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modeling of motions for shallow water with free surface in a rotating sub-domain Marche (2007) [19]. The global existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuum. Unlike the previous analysis about the compressible fluid model without Coriolis forces, see for instance Danchin (2000) [10], Haspot (2009) [16], the rotating effect causes a coupling between two parts of Hodge's decomposition of the velocity vector field, and additional regularity is required in order to carry out the Friedrichs' regularization and compactness arguments.
21. Global existence for compressible Navier–Stokes–Poisson equations in three and higher dimensions (with Hai-Liang Li)
    Journal of Differential Equations 246(12), 4791–4812, 2009.      
    The compressible Navier–Stokes–Poisson system is concerned in the present paper, and the global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces in three and higher dimensions.



■ Well-posedness of IVP/IBVP to dispersive equations

22. Recent progress on some kinds of nonlinear dispersive partial differential equations (in Chinese)
    Research Series for Excellent Doctoral Dissertation of Chinese Academy of Sciences vol. 2007, 1-6, Science Press, Beijing, 2008.
23. Global well posedness for the Gross–Pitaevskii equation with an angular momentum rotational term (with Ling Hsiao, Hai-Liang Li)
    Math. Methods Appl. Sci. 31(6), 655--664, 2008.      
    In this paper, we establish the global well posedness of the Cauchy problem for the Gross–Pitaevskii equation with a rotational angular momentum term in the space $\Bbb{R}^2$.
24. Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions (with Ling Hsiao, Hai-Liang Li)
    J. Math. Phys. 48(10), 102105, 11pp, 2007.      
    In this paper, we establish the global well posedness of the Cauchy problem for the Gross-Pitaevskii equation with an angular momentum rotational term in which the angular velocity is equal to the isotropic trapping frequency in the space $\Bbb{R}^3$.
25. Well-posedness for one-dimensional derivative nonlinear Schrödinger equations
    Commun. Pure Appl. Anal. 2007, 6(4), 997-1021.      
    In this paper, we investigate the one-dimensional derivative nonlinear Schrödinger equations of the form $iu_t-u_{x x}+i\lambda |u|^k u_x=0$ with non-zero $\lambda\in \mathbb R$ and any real number $k\geq 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.
26. Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces (with Ling Hsiao, Baoxiang Wang)
    Journal of Mathematical Analysis and Applications 2007, 328(1), 58-83.      
    We study the well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations $$ i\partial_t u=-\varepsilon\Delta u+\Delta^2 u+P\left((\partial_x^\alpha u)_{|\alpha|\leqslant 2}, (\partial_x^\alpha \bar{u})_{|\alpha|\leqslant 2}\right),\quad t\in \Bbb{R},\ x\in\Bbb{R}^n, $$ where $\varepsilon\in\{-1,0,1\}$, $n\gs 2$ denotes the spatial dimension and $P(\cdot)$ is a polynomial excluding constant and linear terms.
27. Wellposedness for the fourth order nonlinear Schrödinger equations (with Ling Hsiao, Baoxiang Wang)
    Journal of Mathematical Analysis and Applications 2006, 320(1), 246-265.      
    We study the local smoothing effects and wellposedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in 1D $$ i\partial_t u=\partial_x^4 u+P\left((\partial_x^\alpha u)_{\abs{\alpha}\leqslant 2}, (\partial_x^\alpha \bar{u})_{|\alpha|\leqslant 2}\right),\quad t,x\in \Bbb{R}, $$ where $P(\cdot)$ is a polynomial excluding constant and linear terms.
28. Energy scattering theory for the nonlinear Schrödinger equations with exponential growth in lower spatial dimensions (with Baoxiang Wang, Henryk Hudzik)
    Journal of Differential Equations 2006, 228(1), 311--338.      
    For one and two spatial dimensions, we show the existence of the scattering operators for the nonlinear Schrödinger equation with exponential nonlinearity in the whole energy spaces.
29. The initial boundary value problem for quasi-linear Schrödinger-Poisson equations
    Acta Math. Sci. Ser. B Engl. Ed. 2006, 26(1), 115-124.      
    In this article, the author studies the initial-(Dirichlet) boundary problem for a high-field version of the Schrödinger-Poisson equations, which include a nonlinear term in the Poisson equation corresponding to a field-dependent dielectric constant and an effective potential in the Schrödinger equations on the unit cube. A global existence and uniqueness is established for a solution to this problem.
30. Studies on Schrödinger-Poisson systems (Excerpt of Dissertation) (with Ling Hsiao)
    J. Grad. Sch. of CAS 22(5), 141-146, 2005.   
    The bipolar(defocusing nonlinear)Schrödinger-Poisson system and quasi—linear Schrödinger—Poisson equations ale studied．The wellposedness，large time behavior and modified scattering theory is established for the Cauchy problem to the bipolar(defocusing nonlinear)Schrödinger—Poisson systems．The initial-(Dirichlet) boundary problem for a high field version of the Schrödinger-Poisson equations，quasi-linear Schrödinger-Poisson equations，which include a nonlinear term in the Poisson equation corresponding to a field-dependent dielectric constant and an effective potential in the Schrödinger equations on the unit cube ale also discussed．
31. On the initial value problem for the bipolar Schrödinger-Poisson systems (with Hai-Liang Li)
    J. Partial Diff. Eqns. 17(3), 283-288, 2004.   
    In this paper，we prove the existence and uniqueness of global solutions in $H^s(\Bbb{R}^3)$ ($s\in \Bbb{R}，s\geqslant 0$)for the initial value problem of the bipolar Schrödinger-Poisson systems.
32. Modified scattering for bipolar nonlinear Schrödinger-Poisson equations (with Ling Hsiao, Hai-Liang Li)
    Math. Models Methods Appl. Sci. 2004, 14(10), 1481-1494.      
    In this paper, we study the asymptotic behavior in time and the existence of the modified scattering operator of the globally defined smooth solutions to the Cauchy problem for the bipolar nonlinear Schrödinger–Poisson equations with small data in the space $\Bbb{R}^3$.
33. Large time behavior and global existence of solution to the bipolar defocusing nonlinear Schrödinger-Poisson system (with Ling Hsiao)
    Quart. Appl. Math. 2004, 62(4), 701-710.      
    In this paper, we study the large time behavior and the existence of globally defined smooth solutions to the Cauchy problem for the bipolar defocusing nonlinear Schrödinger-Poisson system in the space ${\mathbb{R}^{3}}$.
34. Energy scattering for the generalized Davey-Stewartson equations
    Acta. Math. Appl. Sinica 19(2), 333-340, 2003.      
    Considering the generalized Davey-Stewartson equation $i\mathop u\limits^. - \Delta u + \lambda \left| u \right|^p u + \mu E\left( {\left| u \right|^q } \right)\left| u \right|^{q - 2} u = 0$ where $\lambda > 0,\mu \ge 0,E ={\mathcal {F}}^{ - 1} \left( {\xi _1^2 /\left| \xi \right|^2 } \right){\mathcal{F}}$ we obtain the existence of scattering operator in $\Sigma(\Bbb{R}^n):=\{u\in H^1(\Bbb{R}^n: |x|u\in L^2(\Bbb{R}^n)\}$.
35. Energy scattering for nonlinear Davey-Stewartson equations (in Chinese)
    J. Math. Res. Expo. 23(4), 645-650, 2003.   
    In this paper, the author studies the scattering theory of a class of nonlinear Davey-Stewartson equations with three nonlinearities. The author proves that their scattering operator exists in the energy space $H^1$.
36. Energy scattering for the generalized Davey-Stewartson equations (Rapid Communications)
    Hebei Uni., Rat. Sci. Edt. 22(1), 73-74, 2002.



■ Well-posedness or asymptotic of IVP/IBVP to quantum models

37. Long-time self-similar asymptotic of the macroscopic quantum models (with Hai-Liang Li, Guojing Zhang, Min Zhang)
    J. Math. Phys. 49(7), 073503, 14pp, 2008.      
    The unipolar and bipolar macroscopic quantum models derived recently, for instance, in the area of charge transport are considered in spatial one-dimensional whole space in the present paper. These models consist of nonlinear fourth-order parabolic equation for unipolar case or coupled nonlinear fourth-order parabolic system for bipolar case. We show for the first time the self-similarity property of the macroscopic quantum models in large time. Namely, we show that there exists a unique global strong solution with strictly positive density to the initial value problem of the macroscopic quantum models which tends to a self-similar wave (which is not the exact solution of the models) in large time at an algebraic time-decay rate.
38. Quantum Euler-Poisson system: local existence of solutions (with Yueling Jia, Hai-Liang Li)
    J. Partial Diff. Eqns. 16(4), 306-320, 2003.   
    The one-dim ensional tran sient quantum Euler-Poisson system for sem i-conductors iS studied in a bounded interva1．The quan tum correction can be interpreted as a dispersive regularization of the classical hydrodynamic equations and mechanical efects． The existence an d uniqueness of local-in-time solutions are proved with lower regularity an d without the restriction on the smallness of velocity，where the pressure—density is general(can be non—convex or non-monotone)．



Published Books

39. Harmonic Analysis Method for Nonlinear Evolution Equations,I (with Baoxiang Wang, Zhaohui Huo, Zihua Guo)
    World Scientific Pub Co Inc 300pp, 2011.        
    This monograph provides a comprehensive overview on a class of nonlinear evolution equations, such as nonlinear Schrödinger equations, nonlinear Klein–Gordon equations, KdV equations as well as Navier–Stokes equations and Boltzmann equations. The global wellposedness to the Cauchy problem for those equations is systematically studied by using the harmonic analysis methods.
    This book is self-contained and may also be used as an advanced textbook by graduate students in analysis and PDE subjects and even ambitious undergraduate students.
    @book {WHHG11book,
    AUTHOR = {Wang, Baoxiang and Huo, Zhaohui and Hao, Chengchun and Guo, Zihua},
    TITLE = {Harmonic analysis method for nonlinear evolution equations. {I}},
    PUBLISHER = {World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ},
    YEAR = {2011},
    PAGES = {xiv+283},
    ISBN = {978-981-4360-73-9; 981-4360-73-2},
    MRCLASS = {35-02 (35B30 35Q20 35Q30 35Q55 42-02)},
    MRNUMBER = {2848761},
    DOI = {10.1142/9789814360746},
    URL = {https://doi.org/10.1142/9789814360746},
    }